Use meet when encountering a person for the first time:
- Nice to meet you!
- There’s someone I’d like you to meet.
- I met my best friend in first grade.
- We met some interesting people on our trip.
Use meet with for spending time with someone for discussion (business, politics, etc.)
- I met with Cathy to plan the office Christmas party.
- The president is going to meet with the prime minister on Friday.
Note that meet with must be followed by a person (Cathy, the prime minister).
If both people are listed before the verb, then we use only “meet” – but it still means spending time with someone for discussion, not meeting them for the first time:
- Cathy and I met to plan the office Christmas party.
- The president and the prime minister are going to meet.
MEET UP / MEET UP WITH
These are informal ways to say “get together” for socializing:
- Let’s all meet up in the park at 2:00.
- I met up with my friends after work.
In the first sentence, you could also say “Let’s meet in the park…” but it’s very common to use “meet up” when speaking informally. In the second sentence, “meet up” is followed by people (“my friends”) so that’s why we need the word “with.”